Math of Global Cities

Earlier this week, Cornell mathematician Steven Strogatz reprised George
Zipf's famous power law for the size distribution of cities where "the
population of a city is, to a good approximation, inversely
proportional to its rank."

Tim Gulden of George Mason University has used data based on
satellite images of the world at night to provide a Zipfian analysis
for the 1000 largest urban agglomerations globally, based on several
different measures of city size comparing rankings for
population, economic activity, and patented innovations.

Gulden's research provides what is perhaps the first systematic depiction of the size distribution of global cities, and generates important insights into geographic organization of economic activity and innovation across the world.

While the population of cities tends to follow the Zipf law
that Strogatz describes within a nation, this scaling does not hold for the whole collection of world cities. The distribution ends
up being somewhat flatter - particularly among the largest cities. This
may result from barriers to migration between countries.

The distribution of economic activity - which Gulden estimates using the light emissions from night-time satellite images - is more Zipf-like..

Innovation, however, appears to have a different scaling rule. The
slope of -1.45 indicates that the distribution of
patent activity is much more skew.

Gulden's work again reminds us that cities and urban agglomerations remain a key facet of globalization. As barriers come down and global forces continue to act on cities, the world's cities are likely to eventually conform to the basic rank-size pattern Zipf identified - already evident in the global distribution of economic activity and the even more skewed pattern of global innovation.The world's largest urban agglomerations are likely to become even bigger, while second and third tier cities face ever harsher competition.

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